Definition Of Function Research Articles

Positive definite functions on a regular domain

Abstract We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.

A subset of brain regions within adult functional connectivity networks demonstrates high reliability across early development.

The human cerebral cortex contains groups of areas that support sensory, motor, cognitive, and affective functions, often categorized into functional networks. These networks show stronger internal and weaker external functional connectivity (FC), with FC profiles more similar within the same network. Previous studies have shown these networks develop from nascent forms before birth to their mature, adult-like structures in childhood. However, these analyses often rely on adult functional network definitions. This study assesses the potential misidentification of infant functional networks when using adult models and explores the consequences and possible solutions to this problem. Our findings suggest that although adult networks only marginally describe infant FC organization better than chance, misidentification is primarily driven by specific areas. Restricting functional networks to areas with adult-like network clustering revealed consistent within-network FC across scans and throughout development. These areas are also near locations with low network identity variability. Our results highlight the implications of using adult networks for infants and offer guidance for selecting and utilizing functional network models based on research questions and scenarios.

Animal emotions and consciousness: a preliminary assessment of researchers' perceptions and biases and prospects for future progress.

Scientists and philosophers have long struggled with the question of whether non-human animals experience emotions or consciousness. Yet, it is unclear where the scientific consensus on these topics lies today. To address this gap, we administered a survey of professional animal behaviour researchers to assess perceptions regarding (i) the taxonomic distribution of emotions and consciousness in non-human animals, (ii) respondents' confidence in this assessment, and (iii) attitudes towards pitfalls and potential for progress when addressing these questions. Respondents (n = 100) ascribe emotionality and consciousness to a broad swath of the animal taxonomy, including non-human primates, other mammals, birds and cephalopods. Respondents' attribution of these phenomena was strongly associated with their confidence in their assessments (R 2 > 0.9), with respondents assuming an absence of emotions and consciousness when they were unsure. We also identify an emergent consensus of the components involved in a functional definition of emotions. Researchers are optimistic that tools either currently exist or will exist in the future to rigorously address these questions (>85%) and that animal behaviour, as a field, should do more to encourage research works on emotions (>70%). We discuss implications for publication bias and future work in this area as well as ethical considerations regarding animal care and use.

Global Path Optimal Planning Method for Autonomous Vehicles Based on Image Feature Extraction

In order to solve the problem of autonomous vehicles driving safely on the road while searching for optimal paths to avoid traffic congestion and obstruction, this paper establishes an optimal path planning model based on image feature extraction. The model consists of four parts: the selection of key turning points, interpolation between turning points, definition of the evaluation function, and global optimal path search. First, based on the map information provided by the network, an image feature detection algorithm was proposed, and the key turning points of the map route were selected and stored in an open table. Then, based on mechanical analysis theory, the cubic interpolation between two key turning points of the optimal path was defined. This allows the car to drive along a continuous curve with a certain arc radius, enabling it to drive smoothly, effectively avoid obstacles, and ensure safety. Then, an evaluation function is defined as needed, and different evaluation functions are specified for different road sections to avoid excessive local deviations in planning and to reduce the impact of known obstacles on the path. Finally, the cost of the search path is calculated based on the evaluation function. According to the principle of minimum cost, a globally optimal heuristic search algorithm is proposed to find the globally optimal path. Experimental results show that the proposed global optimal path planning algorithm has excellent performance, with an average accuracy of 96.71% and a search speed of 2.78 ms. The model not only provides accurate path planning and fast path selection capabilities but also has strong robustness against interference.

Assessment of Nuclear Fusion Reaction Spontaneity via Engineering Thermodynamics.

This work recalls the basic thermodynamics of chemical processes for introducing the evaluation of the nuclear reactions' spontaneity. The application and definition of the thermodynamic state functions of the nuclear processes have been described by focusing on their contribution to the chemical potential. The variation of the nuclear binding potentials involved in a nuclear reaction affects the chemical potential through a modification of the internal energy and of the other state functions. These energy changes are related to the mass defect between reactants and products of the nuclear reaction and are of the order of magnitude of 1 MeV per particle, about six orders of magnitude larger than those of the chemical reactions. In particular, this work assesses the Gibbs free energy change of the fusion reactions by assuming the Qvalue as the nuclear contribution to the chemical potential and by calculating the entropy through the Sackur-Tetrode expression. Then, the role of the entropy in fusion processes was re-examined by demonstrating the previous spontaneity analyses, which assume a perfect gas of DT atoms in the initial state of the fusion reactions, are conservative and lead to assessing more negative ΔG than in the real case (ionized gas). As a final point, this paper examines the thermodynamic spontaneity of exothermic processes with a negative change of entropy and discusses the different thermodynamic spontaneity exhibited by the DT fusion processes when conducted in a controlled or uncontrolled way.

Research on Electric Vehicle Powertrain Systems Based on Digital Twin Technology

As a critical component of electric vehicles, the powertrain has a significant impact on the overall performance of vehicles. In addressing the challenge of lengthy testing cycles, this study develops a para model of the powertrain, utilizing digital twin (DT) technology, thereby establishing a framework for simulation testing of multi-controller intermodulation. We establish functional definition coverage testing by designing specific functional requirement use cases, and we validate the failure mechanism via fault injection use cases. The results indicate that the DT testing platform can effectively simulate the operational interactions among various controllers within the powertrain system. In comparison to traditional field testing, the digital twin-based testing methodology offers enhanced operational efficiency and allows for the examination of testing conditions that are impractical to implement in real vehicles, particularly in the context of fault injection testing, thus facilitating the early detection of potential safety risks within the system. The advancement of this technical solution holds significant practical implications for the future mass production and development of electric vehicles.

Vitali Theorems in Non-Newtonian Sense and Non-Newtonian Measurable Functions

In this paper, we first state the ν−Vitali theorems in the non-Newtonian sense. In the second part, we give the definition of the non-Newtonian measurable function and the relation between ν−measurable and real measurable functions. We also study some basic properties of ν−-measurable functions.

Stokes-based analysis for the estimation of 3D dipolar emission

We provide a general description of the measurement capabilities of systems that probe the 3D state of polarization of light emitted by a dipole or a collection of dipoles. This analysis is based on a generalization of the Stokes parameters for 3D polarization, and its goal is to provide insight into what constitutes a good measurement system under specific circumstances, through the definition of appropriate merit functions. Three cases are considered: the general case of arbitrary states of 3D polarization, the special case of 3D linear full or partial polarization states, and the even more specific case of linear dipoles that wobble with rotational symmetry around a central direction. Note that the latter two cases are of interest in fluorescence microscopy. The analysis presented here is illustrated by applying it to two different approaches used commonly in orientation microscopy: PSF engineering and ratiometric measurements.

The Spectral Gamma Function of gas absorption coefficient and its application to analysis and enhancement of the correlated spectral modeling of radiative transfer in non-uniform gaseous media

A novel Spectral Gamma Function is introduced to address the limitations associated with the “correlated” spectrum assumption in radiative transfer modeling in non-uniform gaseous media. The Spectral Gamma Function combines high-resolution gas absorption spectra with their distribution functions, capturing the “correlated” properties of absorption spectra at different thermodynamic states. This concept is applicable both to narrow bands and the full spectrum, enabling the construction of truly “correlated” models and the calculation of statistical relationships between absorption spectra. The paper presents the definition and properties of the Spectral Gamma Function. The Function 1) allows the construction of sets of truly “correlated” high-resolution spectra from which the “distance” to a real set of gas spectra can be characterized; 2) provides a simple way to evaluate Spearman's rank correlation between spectra in distinct states yielding useful information about the statistical relationship between these spectra; and 3) allows for the approximation of a real set of Line-by-Line (LBL) data as a non-linear combination of “correlated” models called MetaLBL modeling. Additionally, the function may be used to demonstrate the optimality of the Rank Correlated Spectral Line Weighted-sum-of-gray-gases (RC-SLW) Model in the space of “correlated” models, thus demonstrating that the RC-SLW Model represents the limit of accuracy in the family of correlated models. This work represents the first proposal of such a development extending modeling approaches beyond those inherent in the assumption of correlated spectra in the field of gas radiation modeling, offering promising insights for practical applications. Moreover, the MetaLBL modeling strategy is applied in a test case from atmospheric sciences for which models with a user-specified accuracy are constructed.

Existence of Three Positive Solutions for Boundary Value Problem of Fourth Order with Sign-Changing Green’s Function

In this paper, we examine a fourth-order equation that has parameter dependency and boundary conditions in three different places. We prove some of the features of the relevant asymmetric Green’s function and infer its exact form. The resulting solutions are still positive and decreasing functions on the entire interval of the Green’s function definition, and they are concave in a specific subinterval, despite the fact that the function’s sign changes on the square of its definition. The fixed point theorem of Krasnoselskii is the foundation of the existence arguments. Next, using the Leggett–Williams fixed point theorem, it is concluded that there are at least three positive solutions. Lastly, an example is provided, to highlight the primary findings of the manuscript.

Sound transmission through an unbaffled plate

Commonly occurring complex structures can often be modelled as simple finite plates mounted without baffles. Thus, the objective here is to study the sound transmission loss of a simply-supported rectangular finite unbaffled plate impinged upon by an acoustic plane wave. The total pressure jump across the plate surface is expressed as the sum of the pressure jump due to pure diffraction, where the plate normal displacement is zero and the pressure jump due to plate radiation. These pressure jumps are computed using the definition of the free-space Green's function in the wavenumber domain and the Euler equation. Then, the coupled equation of motion is solved for the plate displacement. The method of stationary phase is used to derive the closed-form expression for the improper double integral that appears in the far-field acoustic pressure. Finally, the far-field transmitted power is numerically computed and the sound transmission loss of the unbaffled plate is compared with that of the baffled one.

Piezoelectric energy harvesting from walking motion of a passive biped robot model with flexible legs

Passive dynamic walking biped robots utilize the inherent dynamics of their structure to achieve walking motion without continuous actuation, enhancing energy efficiency and stability. Replacing conventional rigid legs with flexible elastic beams in the simplest passive walker will be associated with undesirable vibrations. Despite the positive effect of the damper in reducing vibrations, the current research has used the piezoelectric energy harvesting with the aim of reducing the wasted energy of the gait cycle and improving its stability range. For this purpose, the electromechanical Lagrange equations are determined by applying Hamilton's principle and the assumed mode method and then, the new definition of the step function is obtained through numerical methods, solving a boundary value problem to establish proper initial conditions for stable walking. Numerical simulations indicate that the piezoelectric energy harvester can create a stable period-one gait cycle by optimizing the length of the piezo layers attached to the undamped elastic legs despite the presence of small vibrations. Additional results are presented in bifurcation diagrams, investigating the effect of important physical and structural parameters such as slope angle, length and thickness of the piezo layer.

Searching for Continuous n-Clusters with Boolean Reasoning

A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper.

A perspective on conditional spectrum-based determination of response statistics of nonlinear systems

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system’s displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

A Single-Cell Atlas of the Substantia Nigra Reveals Therapeutic Effects of Icaritin in a Rat Model of Parkinson's Disease.

Degeneration and death of dopaminergic neurons in the substantia nigra of the midbrain are the main pathological changes in Parkinson's disease (PD); however, the mechanism underlying the selective vulnerability of specific neuronal populations in PD remains unclear. Here, we used single-cell RNA sequencing to identify seven cell clusters, including oligodendrocytes, neurons, astrocytes, oligodendrocyte progenitor cells, microglia, synapse-rich cells (SRCs), and endothelial cells, in the substantia nigra of a rotenone-induced rat model of PD based on marker genes and functional definitions. We found that SRCs were a previously unidentified cell subtype, and the tight interactions between SRCs and other cell populations can be improved by icaritin, which is a flavonoid extracted from Epimedium sagittatum Maxim. and exerts anti-neuroinflammatory, antioxidant, and immune-improving effects in PD. We also demonstrated that icaritin bound with transcription factors of SRCs, and icaritin application modulated synaptic characterization of SRCs, neuroinflammation, oxidative stress, and survival of dopaminergic neurons, and improved abnormal energy metabolism, amino acid metabolism, and phospholipase D metabolism of astrocytes in the substantia nigra of rats with PD. Moreover, icaritin supplementation also promotes the recovery of the physiological homeostasis of the other cell clusters to delay the pathogenesis of PD. These data uncovered previously unknown cellular diversity in a rat model of Parkinson's disease and provide insights into the promising therapeutic potential of icaritin in PD.

Типология линериализации прилагательных (на материале английского и кабардино-черкесского языков)

The article discusses the linerialization of adjectives in a typological aspect. The approach to the arrangement of adjectives in multi-system languages is not accidental. The study tests the hypoth-esis that native speakers, regardless of linguistic orientation, trigger the same or similar thought processes at the cognitive level. Examples from English and Kabardino-Circassian languages are chosen as contrasting ones. The issues raised in the paper are of interest both from a practical and theoretical point of view. For example, it is known what difficulties linguists have to face inter-preting nouns used in the definition function, which affects the work of lexicographers and teach-ers engaged in educational activities. The researchers determined during the research that the Eng-lish language has a unified position regarding how the hierarchy of prepositive adjectives in syn-tagmatics is built. An analysis of examples from the Kabardino-Circassian language shows that their attribution influences the distribution of adjectives to different classes – relative or qualita-tive. Relative adjectives are located to the left of the vertex word (preposition), and qualitative adjectives are located to the right (postposition). In general, the idea is confirmed that the lineari-zation of adjectives is universal, while preserving a certain specificity for each language.

Nonparametric and Continuous Variable-Based Stratigraphic Modelling from Sparse Boreholes using Signed Distance Function and Bayesian Compressive Sensing

An accurate stochastic interpretation of subsurface stratigraphy with quantified uncertainty can benefit the subsequent risk management of geotechnical infrastructure. Traditional approaches to developing geological cross-sections from sparse boreholes typically require the calibration or definition of empirical model parameters and functions, which may introduce subjectivity and bias. In this study, a non-parametric and continuous variable-based spatial predictor that leverages the signed distance function and Bayesian compressive sensing (BCS) is proposed for subsurface stratigraphic modelling. The proposed method transforms sparse categorical borehole data from a low-dimensional space into continuous variables in a high-dimensional space, enabling a comprehensive representation of more implicit characteristics of intricate geological patterns. This transformation facilitates the use of the continuous-variable-based BCS for non-parametric spatial prediction. The most probable geological cross-section and uncertainty qualification plot are derived after transforming spatially interpreted fields of continuous variables back into soil types. The performance of the proposed method is demonstrated using synthetic and real-world cases. Results indicate that the proposed approach can handle intricate stratigraphic scenarios characterized by complex geological structures, such as crossed-inclined, folded, inclined-folded, and interbedded strata, in a data-driven and non-parametric manner. The advantages of the proposed method over existing spatial predictors for developing geological cross-sections are also demonstrated.

Efficient Data Exchange between WebAssembly Modules

In the past two decades, there has been a noticeable decoupling of machines and operating systems. In this context, WebAssembly has emerged as a promising alternative to traditional virtual machines. Originally designed for execution in web browsers, it has expanded to executing code in restricted and secure environments, and it stands out for its rapid startup, small footprint, and portability. However, WebAssembly presents significant challenges in data transfer and the management of interactions with the module. Its specification requires each module to have its own memory, resulting in a “share-nothing” architecture. This restriction, combined with the limitations of importing and exporting functions that only handle numerical values, and the absence of an application binary interface (ABI) for sharing more complex data structures, leads to efficiency problems; these are exacerbated by the variety of programming languages that can be compiled and executed in the same environment. To address this inefficiency, the Karmem was designed and developed. It includes a new interface description language (IDL) aimed at facilitating the definition of data, functions, and relationships between modules. Additionally, a proprietary protocol—an optimized ABI for efficient data reading and minimal decoding cost—was created. A code generator capable of producing code for various programming languages was also conceived, ensuring harmonious interaction with the ABI and the foreign function interface. Finally, the compact runtime of Karmem, built atop a WebAssembly runtime, enables communication between modules and shared memory. Results of the experiments conducted show that Karmem represents an innovation in data communication for WASM in multiple environments and demonstrates the ability to overcome challenges of efficiency and interoperability.

Estimation of AMOC transition probabilities using a machine learning based rare-event algorithm

Abstract The Atlantic Meridional Overturning Circulation (AMOC) is an important component of the global climate, known to be a tipping element, as it could collapse under global warming. The main objective of this study is to compute the probability that the AMOC collapses within a specified time window, using a rare-event algorithm called Trajectory-Adaptive Multilevel Splitting (TAMS). However, the efficiency and accuracy of TAMS depend on the choice of the score function. Although the definition of the optimal score function, called “committor function” is known, it is impossible in general to compute it a priori. Here, we combine TAMS with a Next-Generation Reservoir Computing technique that estimates the committor function from the data generated by the rare-event algorithm. We test this technique in a stochastic box model of the AMOC for which two types of transition exist, the so-called F(ast)-transitions and S(low)-transitions. Results for the F-transtions compare favorably with those in the literature where a physically-informed score function was used. We show that coupling a rare-event algorithm with machine learning allows for a correct estimation of transition probabilities, transition times, and even transition paths for a wide range of model parameters. We then extend these results to the more difficult problem of S-transitions in the same model. In both cases of F-transitions and S-transitions, we also show how the Next-Generation Reservoir Computing technique can be interpreted to retrieve an analytical estimate of the committor function.